Rolle's theorem on a quartic

We have $P(0)=2$ and $P(1)=3-11+12-4+2=2$. So $P(0)=P(1)$.

$P$ is a polynomial, hence continuous on $[0,1]$ and differentiable on $(0,1)$. By Rolle's theorem there exists $c\in(0,1)$ with $P'(c)=0$.

Remark. We found the existence of a root of $P'$ without being able to compute it: $P'(x)=12x^3-33x^2+24x-4$ is a cubic with no pleasant root. Rolle's theorem is an *existence* theorem.